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ACCELERATED RUNGE-KUTTA METHODS
AND NON-SLIP ROLLING
by
Artin Farahani
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirement for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
December 2010
Copyright 2010 Artin Farahani

Presented here are the new Accelerated Runge-Kutta (ARK) numerical integration methods for the solution of initial value problems described by ordinary differential equations. A thorough analysis of these methods is given that includes motivation, derivation, parameter selection, accuracy, speed, stability and convergence studies. Also presented, is a new methodology for modeling the dynamics of non-slip rolling of a spherical ball and the path traced on an arbitrary smooth surface along with a comparison of simulation and experimental results.; This study begins with the constant step size autonomous ARK methods. These methods are derived using simple mathematical tools, and they benefit from a precise parameter selection technique that increases their order of accuracy for some problems. These constant step size ARK methods are proved to be convergent and stable given certain conditions, and their stability regions are presented. Numerical examples show that in general for a small increase in execution time, the 3rd and 4th order ARK methods give solutions that are many times more accurate than those of the 2nd and 3rd order Runge-Kutta methods, respectively.; These constant step size methods are extended to the more general non-autonomous form, which can solve non-autonomous problems directly. The conditions of stability and convergence as well as the stability regions of these methods are the same as those of autonomous ARK methods. These methods are as accurate yet faster than the autonomous ARK methods.; The main result of this study, the new variable step size method, ARK34, which controls the integration error through automatic step size selection is presented. Variable step size ARK methods are proved to be stable and convergent given certain conditions, and their stability regions are also presented. An interpolation scheme is developed for ARK34 that allows it to give solutions at any desired time. Numerical experiments show that the ARK34 solver can be simultaneously as much as ten times more accurate and up to four times faster than the ODE23 solver of Matlab. ARK34 can even be ten times more accurate than the ODE45 solver of Matlab (although not as fast).; A new method is developed for modeling complex nonlinear mechanical systems. This method allows development of the equations of motion for non-slip rolling of a spherical ball on an arbitrary smooth surface and finding the path of motion traced by the point of contact. A novel approach is used to model the surface traced path through a massless point that is included in the system coordinates. This approach utilizes the fundamental equation in a systematic way and produces the path directly upon solution of the equations of motion. The methodology is applied to the motion of a spherical ball rolling non-slip on a cylindrical surface. Simulations and experiments are performed and the results are compared. Sensitivity to initial conditions appear to in large measure explain the difference seen between simulations and experiments.

ACCELERATED RUNGE-KUTTA METHODS
AND NON-SLIP ROLLING
by
Artin Farahani
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirement for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
December 2010
Copyright 2010 Artin Farahani